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We consider homogenization problems for linear elliptic equations indivergence form. The coecients are assumed to be a local perturbation of someperiodic background. We prove $W^{1,p}$ and Lipschitz convergence of thetwo-scale expansion, with explicit rates. For this purpose, we use a correctoradapted to this particular setting, and dened in [10, 11], and apply the samestrategy of proof as Avellaneda and Lin in [1]. We also propose an abstractsetting generalizing our particular assumptions for which the same estimateshold.

Precised approximations in elliptic homogenization beyond the periodic setting